4.8 Multiple Integrals and Monte Carlo Integration

• Composite Trapezoidal rule to approximate double

Integral: 𝑓 𝑥, 𝑦 𝑑𝐴𝑅 where 𝑅 = 𝑥, 𝑦 𝑎 ≤ 𝑥 ≤

𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑} for some constants 𝑎, 𝑏, 𝑐 and 𝑑.

• Let 𝑘 = (𝑑 − 𝑐)/2, ℎ = (𝑏 − 𝑎)/2.

• 𝑓 𝑥, 𝑦 𝑑𝐴𝑅= 𝑓 𝑥, 𝑦 𝑑𝑦

𝑑

𝑐

𝑏

𝑎𝑑𝑥

≈ 𝑑 − 𝑐

4[𝑓 𝑥, 𝑐 + 2𝑓 𝑥,

𝑐 + 𝑑

2+ 𝑓(𝑥, 𝑑)]

𝑏

𝑎

𝑑𝑥

≈ 𝑑 − 𝑐

4[𝑓 𝑥, 𝑐 + 2𝑓 𝑥,

𝑐 + 𝑑

2+ 𝑓(𝑥, 𝑑)]

𝑏

𝑎

𝑑𝑥

=(𝑏 − 𝑎)

4

(𝑑 − 𝑐)

4[𝑓 𝑎, 𝑐 + 2𝑓 𝑎,

𝑐 + 𝑑

2+ 𝑓(𝑎, 𝑑)]

+(𝑏 − 𝑎)

4

(𝑑 − 𝑐)

42 𝑓𝑎 + 𝑏

2, 𝑐 + 2𝑓

𝑎 + 𝑏

2,𝑐 + 𝑑

2+ 𝑓𝑎 + 𝑏

2, 𝑑

+(𝑏 − 𝑎)

4

(𝑑 − 𝑐)

4[𝑓 𝑏, 𝑐 + 2𝑓 𝑏,

𝑐 + 𝑑

2+ 𝑓(𝑏, 𝑑)]

The approximation is of order 𝑂((𝑏 − 𝑎)(𝑑 − 𝑐)[ 𝑏 − 𝑎 2 + (𝑑 − 𝑐)2])

Monte Carlo Simulation

The needle crosses a line if 𝑦 ≤ 𝐿/2sin(𝜃) Q: What’s the probability 𝑝 that the needle will intersect on of these lines? • Let 𝑦 be the distance between the needle’s midpoint and the

closest line, and 𝜃 be the angle of the needle to the horizontal. • Assume that 𝑦 takes uniformly distributed values between 0 and

D/2; and 𝜃 takes uniformly distributed values between 0 and 𝜋.

• Let 𝐿 = 𝐷 = 1.

• The probability is the ratio of the area of the shaded region to the area of rectangle.

• 𝑝 = 1

2𝑠𝑖𝑛𝜃𝑑𝜃𝜋0

𝜋/2= 2/𝜋

Hit and miss method: The volume of the external region is 𝑉𝑒 and the fraction of hits is 𝑓ℎ. Then the volume of the region to be integrated is 𝑉 = 𝑉𝑒𝑓ℎ.

Algorithm of Monte Carlo

• Define a domain of possible inputs.

• Generate inputs randomly from a probability distribution over the domain.

• Perform a deterministic computation on the inputs.

• aggregate the results from all deterministic computation.

Monte Carlo Integration (sampling)

𝐴 = lim𝑁→∞ 𝑓(𝑥𝑖)∆𝑥

𝑁

𝑖=1

Where ∆𝑥 =𝑏−𝑎

𝑁, 𝑥𝑖 = 𝑎 + (𝑖 − 0.5)∆𝑥.

𝐴 ≈𝑏 − 𝑎

𝑁 𝑓(𝑥𝑖)

𝑁

𝑖=1

Which can be interpreted as taking the average over 𝑓 in the

interval, i.e., 𝐴 ≈ 𝑏 − 𝑎 < 𝑓 >, where < 𝑓 >= 𝑓 𝑥𝑖𝑁𝑖=1

𝑁.